Exact Bootstrap Solutions to the Low Equation

Abstract

We inquire whether the Low equation for meson-baryon scattering in the static one-meson approximation possesses bootstrap solutions, defined to be those solutions satisfying Levinson's theorem of potential scattering. The Low equation is allowed to have arbitrary subtractions and arbitrary bound states, except that the baryon must be the lowest bound state. A two-parameter family of cutoff functions, including the case of no cutoff, is introduced. We consider 2\ifmmode\times\else\texttimes\fi{}2 and 4\ifmmode\times\else\texttimes\fi{}4 crossing matrices of certain general forms, with the Chew-Low theory included as a special case of the latter. To answer the question raised, a simple technique is used based on the crossing relation on the imaginary energy axis. It is shown that for all the crossing matrices considered the unsubtracted Low equation does not possess a bootstrap solution, regardless of the choice of bound states and cutoff function. We further study the case of one subtraction for 2\ifmmode\times\else\texttimes\fi{}2 crossing matrices and find some necessary bootstrap conditions. These restrict the crossing matrix, determine the form of the cutoff function, and require that in the baryon channel the baryon be the only bound state, while in the other channel there be at most one bound state. These results are generalizations of those obtained earlier by Huang and Low.